Two-Dimensional Dam Break Flow

Theresa Henke, REU Student

Amanda L. Wood, Mentor

Dr. K. H. Wang, Faculty Mentor

Department of Civil and Environmental Engineering

University of Houston

Houston, TX

he research study described herein was sponsored by the National Science Foundation under the

Award No. EEC-0649163. The opinions expressed in this study are those of the authors and do not

necessarily reflect the views of the sponsor.”

2

Table of Contents

Abstract…………………………………………………………………………….……………3

Introduction…………………………………………………………………………………..4

Background theory………………………………………………………………..……...8

Experimental Study……...…………………………………….………………………..13

Experimental Results…………………………………………….………………….….19

Discussion of Experimental Results.……………………….………………….….31

Numerical Results…………………………………………………………………………33

Discussion of Numerical Results…………………………………………………..34

Conclusion…………………………………………………………………………….……..34

References………………………………………………………….…………….……..….35

Appendix………………………………………………………………………………..……36

3

Abstract

The contents of this paper outline an experiment representing dam break flow in an

open, dry bed channel containing a 90° bend. The dam break experiment was designed to test

a two-dimensional numerical model based off a system of shallow water partial differential

equations known as the St. Venant equations. Experiments were performed and

measurements were taken with both wave gauges and video to verify the water depth and

associated time predicted by the numerical model. The data recorded accurately depicts the

observed behavior of the flow that occurred during the experiment.

4

Introduction

Throughout the years great efforts to accurately model the propagation of flood waves

from dam breaks have been made. Flood waves can have an extremely powerful and

dangerous effect on inhabitants and the environment of the area surrounding the dam.

Therefore, it is important to be able to model a dam break wave as it travels out from the dam

so that authorities will know which areas are most critical to alert. In order to model these dam

breaks many methods have been employed. Such methods include various different analytical

and numerical schemes. Many of these schemes are based off a very important concept in

open channel flow, the St. Venant equations. The St. Venant equations are a system of

hyperbolic partial differential equations modeling continuity and momentum. Continuity and

momentum are important in open channel flow because they both must be conserved in the

closed system. Thus, the St. Venant equations govern unsteady flow in an open channel. These

equations will be discussed more in depth later. The models that are produced from these

schemes based off the St. Venant equations need to be verified with experimental data in order

to ensure they are correct because no explicit solutions to the St. Venant equations exist. The

purpose of this paper is to outline an experiment that was used to test a numerical model that

was produced to predict the propagation of a wave resulting from a complete dam break in a

two-dimensional channel with a 90° bend.

The experiment described in this paper involves a complete dam breach simulated by

the removal of a sluice gate holding back a reservoir of water. The downstream channel is

rectangular and contains a 90° bend. Both wave gauges and a video camera were used to

record water depths versus time. Other similar experiments have been performed by others in

order to ensure the accuracy of their numerical models as well. One such experiment was

performed with similar test conditions by Frazão et al. (2002). In their experiment an upstream

reservoir was breached into a dry glass prismatic channel which had a 90° bend in it after 4m

and then continued on another 3m until it emptied out. The flow was imaged by high-speed

digital cameras both above and alongside the channel. Particle-tracking-velocimetry, which

tracks small floaters in the water, was also used to gather information about flow velocities.

5

This experiment was performed on a larger scale and used more techniques to acquire data on

the wave propagation.

Another similar experiment performed in a channel with a bend in it was performed by

Miller et al. (1989). This experiment involved the breaching of an upstream reservoir into a

channel with two straight sections connected by a 180° bend. Like our channel it was made

from clear materials as well Plexiglas and glass so that cameras could shoot the water as it

moved through the channel. This experiment was quite larger than our experiment with and

upstream channel having dimensions of 3.65m wide, 2.3m long and 0.4m deep. Miller et al.

(1989) also used capacitance probes to measure water levels but only in the reservoir because

they found the probes unreliable in measuring such varied, unsteady flow. As a result, three

video cameras were placed along the channel sides. Like in our experiments an electronic clock

was placed in view of the camera along with a tape measure on the channel side. The clock

allowed them to know the exact time associated with the water height in each of the picture

frames. This experiment also only had one camera available to take recordings and thus had to

be performed many times.

Regarding the issue related to the measurement of water depth in a dam break test, a

completely different, more complex method used to record water depths used by Eaket et al.

(2005) is Stereoscopy. Their experiment was set up in a large tank divided in half by a wooden

dam as opposed to being performed in a channel. Water behind the dam had floating particles

placed on the surface. Above the tank three time-synchronized video cameras were positioned

so that at all times at least two of the cameras would be photographing the below tank. Once

the dam was breached velocities of the particles were determined based on the translation of

particles between successive image sets. A computer algorithm was then used to interpret the

stereo images of the tracking particles to obtain x, y, and z coordinates. This method, however,

was only good for dam breaks on to a dry bed because the particles became entrained in the

water when wet bed experiments were performed.

Aureli et al. (2008) conducted dam break experiments in a similar type tank with a

camera filming technique. Instead of using floating particles to measure flow velocities and

6

depths grayscale images were taken and converted to depths. This was done by complete

blacking out the room and dying the water. The calibration of grey tone to corresponding

water depth was performed by filling the tank up to different water levels and taking pictures

and noting what grey tones corresponded to what water depths. The result of using this

method to capture the propagation of a wave from a dam break is that spatial data about water

depths for the entire area of interest could be recorded all at once. This is much more

thorough than only a few point measurements.

Other efforts have been made to model a dam break in a more realistic situation. The

experiment performed by Nsom (2002) used a viscous fluid to test the effects of mud and other

debris in the flow which make flow more viscous. A horizontal channel, 5m long, 0.3m wide,

and 0.08m high was used and to create a reservoir two Plexiglas plates were placed in the

channel. Flow images from cameras recorded the wave front evolution along 8 different

sections of the straight channel. The camera positioned near the gate was an ultra-fast camera

(1000 images/sec) because this was deemed the most critical spot to record flow. At the other

stages a slower camera was used (25 images/sec). The images were then treated using a

software program. Ultrasonic distance measurements were also taken using a probe which

emitted a sound wave directed at the fluid’s surface. The returning wave could then be

analyzed to give the flow depth. Also, to obtain surface velocity, floating particles are

deposited on the free surface and their velocity was measured by timing them. This

experiment used many different techniques to record data as opposed to the experiment

outlined in this paper which only uses two methods.

Dam break experiments which data were recorded using more than one method was

performed by Stansby et al. (1998). Their experiments were mainly concerned with the initial

features of the wave resulting from a dam break. Therefore, a much larger channel than the

one discussed in this paper was used. A 15.24m long and 0.4m wide flume with a gate placed

at 9.76 meters made the majority of the flume be taken up by the dam. Similar to the

experiment in this paper a thin 3mm thick metal plate was used as the gate which separated

the upstream dam from the downstream channel. A rope was attached to the gate and a pulley

system was used to lift the gate. One of the techniques unique to this experiment that was

7

used was the recording of flow visualizations by a laser light sheet made by a 4W argon-ion

laser with a fiber-optic cable which directed the beam above the flume through a lens to

produce a vertical light sheet perpendicular to the water’s surface. Flow was also recorded on

a CCD video camera which recorded 25 frames per second. The images were then digitalized

with a computer. However, the camera did not cover the entire area of interest so the cameras

had to be repositioned and the experiment replicated. Velocity fields were also taken using

particle-imaged velocimetry method. Surface elevations at various longitudinal positions were

also measured with resistance probes much like the ones used in the experiment presented in

this paper and confirmed by the video images.

Another important set of experiments were performed in the 1960’s by the Army Corps

of Engineers. Instead of performing their own experiments many researchers have used the

results from the experiments performed by the Corps to validate their numerical models. All

experiments were performed in a 4ft wide, 400ft long, and 21in high wooden flume (slope =

0.005) with a recirculating pump providing the water. The dam was modeled by a gate

positioned 200ft downstream in the flume; behind the gate the water was 1ft deep. The flume

contained one transparent glass sidewall and was lined with 3/8” plastic-coated plywood for

durability. To simulate an instantaneous dam break a hundred pound weight was attached to

the gate and dropped. Once the gate opened it triggered a microswitch to start all cameras and

timers that were connected together on one circuit. Like in the experiment outlined in this

paper the cameras and timers were a means used to record flow depth versus time. A staff rod

was placed alongside the flume wall and a video camera with a clock in its view captured water

height in the flume with respect to time. The video cameras were 16mm and the timers

recorded to a 1/100 of a second.

Thus, the techniques which are employed to perform the experiment outlined in

this paper have been used by many others dating all the way back to the 1960’s and have been

proven effective.

8

Background Theory

In order to accurately simulate open channel flow a numerical model is used to predict

main flooding characteristics, water depths, velocities, and arrival times. A system of non-linear

partial differential equations describing the flow in two-dimensions can be set up. These

differential equations are based off two very important concepts in dam break flow: continuity

and momentum. These two equations are known as the St. Venant equations. The St. Venant

equations were developed by the French mathematician Adhémar Jean Claude Barré de Saint-

Venant (1797-1886). In 1868, at the age of 71, he was elected to the mechanics section of the

Académie des Sciences, where he continued to do research for another 18 years. It was in 1871

that he derived the equations for unsteady flow in open channels. In addition to deriving a

system of equations used in open channel flow, he also derived solutions for the torsion of

noncircular cylinders, the correct derivation of the Navier-Stokes equations for a viscous flow,

and was the first to "properly identify the coefficient of viscosity and its role as a multiplying

factor for the velocity gradients in the flow. He is definitely considered to be one of the great

mathematicians/scientists of his time.

The equations Adhémar Jean Claude Barré de Saint-Venant derived have some

limitations to them however. The St. Venant equations are only appropriate for use in shallow

water flow. Also, in order to apply the St. Venant equations some assumptions must be made.

The first assumption being that the pressure distributions must be hydrostatic; which is a good

assumption to be made if the streamlines do not have sharp curves. The next is the slope of

the channel bottom is very small. Also, uniform velocity across the entire cross section of the

channel must be assumed. Finally, the mass density of the fluid must be constant. With these

assumptions the continuity and momentum equations can be applied to solve problems in open

channels. The continuity equation deals with the conservation of mass. The mass density of

water is assumed to be constant therefore, mass must be conserved in the control volume;

which in this case is the water in the channel. For the case of one-dimensional flow, the

continuity equation is given as

���� + � ��

�� + � ��� = 0 (1)

9

where h is the water depth and v is the averaged velocity. The second concept of conservation

of momentum states that the rate of change of momentum is equal to the resultant force

acting on the control volume. Again, for one-dimensional flow conditions, the momentum

equation is

��� + � �

�� + � ���� = �(�� − ��) (2)

where �� is the bottom slope, �� is the energy slope, and g represents gravitational constant.

However, except for very simplified cases, the St. Venant equations do not possess a closed-

form solution. Originally, before numerical methods were developed, the St. Venant equations

were solved using the method of characteristics which can predict the propagation of a wave in

one-dimension.

Method of Characteristics

According to the method of characteristics the characteristic velocities as given below

can be used to describe the characteristic direction,

� ± � (3)

where the “+” and “-“ signs are for the �� characteristic and �� characteristic, respectively. (Fig.

1) The celerity of a wave is given by equation 4.

� = �� �� (4)

This experiment deals with a rectangular channel which means that the cross-sectional area of

the channel can be described by equation 5.

� = �� (5)

Therefore, � can now be simplified to equation 6.

� = ��� (6)

10

This is important because now we have three equations, continuity, momentum, and celerity,

relating three unknowns: mean water velocity, depth, and celerity. The first step in simplifying

these equations involves substituting � in for � into the continuity and momentum equations,

equations 1 and 2. Next, the equations are then combined and simplified yielding equation 7.

�(±��)�� = �(�� − ��) (7)

The simplified equation, equation 7, can then be integrated to solve the flow velocity and

celerity as shown below

� ± 2� = �(�� − ��) !" (8)

Again, the equation with “+” sign is solved for the solutions along the �� characteristics moving

with speed #�#� = � + � and “-“ is for the solutions along the �� characteristics with

#�#� = � − �.

Two more factors are shown to affect the solutions of characteristic equations; bottom slope,

��, and friction slope, �� . Both of these are assumed to be zero since the channel is flat and

very smooth. The characteristics equation, Eq. 8, can then again be simplified yielding

� ± 2� = �$%&"'%" (9)

Once these characteristic equations are plotted on the x-t plane they become known as

characteristic lines or just characteristics. The two known points, A and B (Fig. 1) represent the

known initial conditions. Where the characteristic lines intersect, point P, the unknown velocity

and celerity at subsequent time can be determined by solving Eqs. 9 algebraically along ��and

c� characteristics.

11

The method of characteristics, however, is not usually used for solving wave

propagation in two-dimensions because it’s limited to one-dimensional flow. Thus, other

numerical methods must be employed in order to obtain approximate solutions by solving two-

dimensional St. Venant equations otherwise known as the shallow water equations for

unsteady flow.

Continuity equation (conservation form):

���� + �()�)

�� + �(�)�� = 0 (10)

Momentum equation along the x-direction:

�()�)�� + �()*�+�.-.�*)

�� + �()�)�� = �ℎ(��� − ���) (11)

Momentum equation along the y-direction:

�(�)�� + �()�)

�� + �(*�+�.-.�*)�� = �ℎ(��� − ���) (12)

Figure 1. Plot of characteristic lines used to

find the solution to the St. Venant Equations

A B

12

Since the numerical model is only an approximation it is necessary to validate numerical

models with results from experiments performed in the laboratory to ensure accuracy. The

experiment this paper outlines deals with shallow, unsteady, open channel flow resulting from

a sudden, complete dam break. In this particular experiment a dry bed is used which, as

demonstrated in Figure 2, results in a smooth curve of water running from the surface of the

water down to the channel bottom. A second type of wave propagation, Figure 3, forms when

a dam is broken onto a wet bed. In this type of dam break a shock or bore is created when the

moving wave of water pushes up the existing water to create a steep wave front moving down

the channel. Thus, the purpose of this experiment is to validate a numerical model which has

been created to predict the propagation of a wave over a dry bed in an open channel.

Figure 2. Wave Propagation over Dry Bed

Bore

Figure 3. Wave Propagation over Wet Bed

13

Experimental Study

Wave Gauge

Wave gauges probes can be used in a number of applications. They include the

measurement of water level, the determination of a liquid's flow rate, and the moisture

content of soil. Both resistance and capacitance probes were used in this experiment to

measure water depth versus time (Fig. 4 & 5). However, the resistance probe was used for the

majority of the data because the capacitance probe could not be positioned in the center of the

channel. The probe works by obtaining an analogic signal from measuring the electric

conductivity of the water as the depth between two parallel probes varies. The probe feeds

data directly to a portable computer or workstation, so analog measurements are immediately

converted to digital records. An oscillator and associated capacitance-resistance network

connected to the probe serve to provide a linear output voltage proportional to the change of

capacitance between the insulated wire and the ground electrode.

Figure 4: Resistance gauge used at locations A,B,C,E,F Figure 5: Capacitance gauge used at location 26

14

Calibration of Gauges

The gauges needed to be calibrated because they output a voltage and height is what

was desired. Therefore, in order to calibrate the gauges they were placed in stationary water of

known depth and then the voltage was recorded (Fig. 6). This produced a set of data relating

voltage and height. This data was then plotted and a linear relationship was determined and

fitted to a straight line (Fig. 7). This equation was then used to determine the height of the

water from the voltage reading given by the gauge.

During the calibration of the gauge it was found that the resistance gauge was unable to

record water depths less than one inch. Therefore, it was determined that for depths recorded

in this experiment that are less than one inch are unreliable.

Figure6: Calibration set-up for resistance gauge

Figure7: Calibration chart for resistance gauge

Resistance Gauge

y = -4.1281x + 26.179

0

1

2

3

4

5

6

7

8

9

10

0 2 4 6 8

Volts

Inch

es

voltage

Linear (voltage)

15

Video Camera

The video camera used in this experiment was a Cannon DC330 (30 frames per second).

The camera was positioned on a tripod aimed at the 90° bend to capture the splash at the

bend. A probe was unable to be used because the water wetted the probes causing them to

function improperly as well as the probes not being long enough. Thus, the video was analyzed

in order to record the wave height versus time. Video editing software was used to clip the

footage so that time began right as the gate opened. The video was then analyzed in slow-

motion to match the corresponding times and heights.

Figure8: Set-up of video camera and spot light at 90° bend

16

Procedure

This experiment is set up and designed to validate a numerical model that has been

created. What is unique to the numerical model and dam break experiment is that they both

contain a 90° bend in the downstream section of the channel. As depicted in Figure 9 the bend

occurs 155in. from the site of the dam break. Also, a square reservoir 35in. by 35in. is placed at

the upstream end of the channel to simulate the dam. The channel and reservoir are separated

by a metal gate connected to a pulley system, Figure 10.

Figure 9: Dimensions of channel and reservoir

17

Figure 10: Pulley system used to lift gate

The pulley system is operated manually with an average opening time of 0.17 seconds.

In order to minimize leakage from the reservoir into the channel and maintain an essentially dry

bed the sides of the gate have small rubber cords inserted along the gate’s sides. In addition,

aluminum tape is placed at the bottom of the gate to minimize water leakage, Figure 11.

Immediately before the gate is to be lifted the rubber cords are removed and the aluminum

tape at the bottom of the gate is cut. Although both of these methods are used to prevent

water from leaking into the channel from the reservoir there is still a small amount of water

that manages to leak out into the channel. The leakage is so small, however, that it cannot

even be measured and thus, its effects are considered negligible. Also, right after the tape is

cut the water height in the reservoir is read from the side of the tank, Figure 12.

18

Figure 11: Gate during filling of reservoir Figure12: Tape measure on side of reservoir

Before the gate is lifted a video camera and spot light are situated at the 90° bend and

turned on to capture the wave as it strikes into the 90° bend in the channel. At the 90° bend

two tape measures, like the ones on the reservoir, are placed so that the height can be

determined from the footage. The gauges are positioned in the channel at various locations,

Figure 14. The flow is too turbulent at the 90° bend and would wet the probe which functions

improperly upon getting wet. Also, the gauge is not long enough to capture the entire height of

the wave. Thus, the gauges were not appropriate for measuring the wave height at the 90°

bend. Instead, a video camera was positioned to capture the wave height versus time, Figure 8.

The height can then be read from the video footage and associated with the time determined

from the clipped footage.

19

Experimental Results

The following results were obtained in July 2009 in the Hydraulics laboratory at the

University of Houston (Fig. 15). Four runs recording water depth versus time were recorded at

gauge locations A,B, and C and 3 runs at locations 26, E, and F, Figure 13. Video verifications

were also taken at locations A, 26, B, and C. The initial reservoir height was approximately 17

inches for all runs to accommodate better comparisons between the graphs. Also, each gauge

was firmly attached directly in the center of the channel to a metal bar so that the propagating

wave would not move the gauge thus affecting the results (Fig 14). Still images have also been

included to verify the behavior of the flow at the gate (Fig. 19), inside the straight section of the

channel (Fig. 20), location A (Fig. 16), Location B, (Fig. 17), Location C (Fig. 18), and the 90° bend

(Figs. 21&22).

Figure 13: Location of wave gauges in channel

20

Figure 14: Set-up of resistance gauge in middle of channel

Figure 15: Set up of Plexiglas channel in laboratory

21

Graph 1: Time series plot of water depth at location A. All initial reservoir heights 17.5”

Location A

0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Time (s)

Wa

ve

He

igh

t (I

n)

A-1

A-2

A-3

A-4

22

Figure16: Video verification at location A

1 2

3 4

5 6

23

Graph 2: Time series plot of water depth at location B. All initial reservoir depths 17”

Figure17: Video verification at location B

Location B

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 2 4 6 8 10 12 14 16 18 20 22 24

Time (s)

Wav

e H

eig

ht

(In

)

B-1

B-2

B-3

B-4

1 2

3 4

24

Graph 3: Time series plot of water depth at location C.

Initial reservoir depths for, C-1& C-2- 17.5”, C-3& C-4- 17”

Location C

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14 16 18 20 22

Time (s)

Wa

ve

Heig

ht

(In

)

C-1 C-3

C-2 C-4

25

Figure 18: Video verification at location C

1

2

3

26

Graph 4: Time series plot of water depth at location E. All initial reservoir depths 17”

Graph 5: Time series plot of water depth at location F. All initial reservoir depths 17”

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Wav

e H

eig

ht

(In

)

Time (s)

Location E

E-1

E-2

E-3

0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Wav

e H

eig

ht

(In

)

Time (s)

Location F

F-1

F-2

F-3

27

Graph 6: Time series plot of wat er depth at location 26. All initial water depths 17”

Graph 7: Time series plot of water depth at 90° bend. Initial water depth 17”

Location 26

0

0.5

1

1.5

2

2.5

3

3.5

4

0 2 4 6 8 10 12 14 16 18 20

Time (s)

Wa

ve

Heig

ht

(In

)

26-1 26-2

26-3

Wall at 90° Bend

0

5

10

15

20

25

0 1 2 3 4 5 6 7 8 9 10

Time (s)

Heig

ht (in)

Wall at 90 Bend

28

Figure19: Flow exiting into the channel from reservoir

Figure 20: (Left) Diamond flow pattern (Right) Wave propagating back into reservoir

1 2

3 4

5 6

29

Figure 21: Flow hitting outside of 90° bend

1 2

3 4

5 6

7 8

30

Figure 22: Wave hitting inside of 90° bend

ERROR: stackunderflow

OFFENDING COMMAND: ~

STACK: